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Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms for $\alpha \in (1/3,1/2)$ are uniformly bounded - that is $\sup_n \|X^n\|_\alpha<\infty$. Define the standard Riemann …

Consider the RDE $$dY^n=b_n(Y^n) \, dt+\sigma(Y^n) \, d\mathbf X$$ where $\mathbf X$ is a rough path, $\sigma$ is as smooth as you'd like and $b_n$ are Lipschitz. If $b_n\to b$ uniformly then Friz-Vic …

In Friz and Hairer's notes on rough paths, there is exercise 2.9 which is called the "interpolation theorem". It says that if you have a sequence of rough paths $\mathbf X^n=(X^n,\mathbb X^n)\in \math …

Consider the rough differential equation $$dY_t=b(Y_t,t) \, dt+\sigma(Y_t,t) \, d\mathbf X_t,$$ where $\mathbf X$ is a $p$-rough path with $1\leq p<3$. If $b$ and $\sigma$ are $C^3_b$ then we have exi …

Let $(X,\mathbb X):[0,1]^2\to \mathbb R^d\oplus\mathbb R^{d\times d}$ satisfy the following four properties: \begin{align} &X_{s,t}=X_{0,t}-X_{0,s}\\ &\sup_{t\neq s}\frac{|X_{s,t}|}{|t-s|^\gamma}<\inf …

My background is in rough paths theory. In short, if you have an irregular function $f:[0,T]\to\mathbb R^d$ and you want to make sense of integrals $\int_s^t \cdot \ df(r)$, the right objects that are …